In this article, we construct ${SL}_k$-friezes using Plücker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Plücker embedding. When this cluster algebra is of finite type, the ${SL}_k$-friezes are in bijection with the so-called mesh friezes of the corresponding Grassmannian cluster category. These are collections of positive integers on the AR-quiver of the category with relations inherited from the mesh relations on the category. In these finite type cases, many of the ${SL}_k$-friezes arise from specializing a cluster to 1. These are called unitary. We use Iyama–Yoshino reduction to analyze the nonunitary friezes. With this, we provide an explanation for all known friezes of this kind. An appendix by Cuntz and Plamondon proves that there are 868 friezes of type $E_6$.

在本文中，我们构造 ${SL}_{ķ}$-利用普吕克坐标进行带状装饰，利用聚类结构在格拉斯曼方程的齐次坐标环上 $ķ$中的空格 $ñ$-通过Plücker嵌入实现空间。当该簇代数为有限类型时，${SL}_{ķ}$-条带与相应的格拉斯曼聚类类别的所谓网状条带双射。这些是类别的AR颤抖上正整数的集合，具有从类别的网格关系继承的关系。在这些有限类型的情况下，许多${SL}_{ķ}$-friezes是将集群专门化为1产生的。这些被称为单一。我们使用Iyama–Yoshino归约法来分析非单一fr带。借此，我们对所有这种已知的fr带进行了说明。Cuntz和Plamondon的附录证明，有868种类型的fr带${Ë}_6$。